Of course the "rivals" aren't always enemies. Usually they can be friendly rivals or even siblings but make sure the elements are ones that would be in natural opposition before you add an example. (Rock and Dirt, for example are too similar)

In Naruto, the titular hero has Wind chakra, while Sasuke has Fire (beats Wind) and Lightning (loses to wind) Chakra. Another friendlier example could be Kisame and Itachi (who can use Fire techniques.)

In Gamaran, the Boisterous Bruiser Zenmaru specialize in the "Kagutsuchi Kata" (Fire God Form) style, while the much calmer Shinnojo is an expert of the "Mizuchi Kata" (Water Dragon Form).

In Fairy Tail there are Natsu (Fire) and Gray (Ice). Another elemental-related rivalry is the one of Makarov (Light) and Jose (Darkness).

Digimon Savers: In their burst mode evolutions Masaru powers up by summoning a miniature sun and dropping it on himself, while Thoma powers up by summoning a crescent moon.

In Rurouni Kenshin: Kenshin and the Kyoto arc Big Bad Shishio Makoto can be considered rivals (both were highly skilled hitokiri during the Bakumatsu period (1853-1868), and now they're enemies. In their final battle, Kenshin's revamp and secret attack summons forth a fierce squall, while Shishio's special attack and power up are announced by roaring flames which ultimately lead to his Karmic Death as he eventually overheats and burns alive.

Rin Natsuki/Cure Rouge and Karen Minazuki/Cure Aqua from Yes! Pretty Cure 5. The former uses fire elemental magic and is designated as the Cure of Passion. The latter uses water elemental magic and is designated as the Cure of Intelligence. These two characters early on are depicted as rivals that couldn't get along most of the time and hate being one-upped by the other. However, over the first season, they came to an understanding, and the rivalry became less severe.

Samurai Deeper Kyo has the fire user Hotaru, red oni to his half-brother Shinrei's blue oni, who can manipulate water. The two also sports opposite behaviours and contrasting hair colors (blonde for Hotaru, silver for Shinrei). During his days as a member of the Four Saints he also had a bit of rivalry with Akira, who can use ice powers.

Comic Book

Namor the Sub-Mariner and the original Human Torch. Namor does not actually have water powers but is at his strongest underwater and while operating on the surface will periodically take dips to restore himself. The Human Torch is a fire shooting android that defends New York while Namor seeks to destroy it.

X-Men has Surge and Dust who are rivals. They are only ideological rivals though as Dust is at a severe disadvantage in a physical contest. Another example is given by Iceman, a shape shifter who draws energy from his surroundings and Havok, whose laser beam shooting body requires a special suit to keep under control. They are a bit more evenly matched but, again, are romantic rivals for a woman's affections more than anything else.

Batman and the Outsiders has a villainous couple with heat and cold powers, who are actually trying to find a cure to their opposite powers.

Iceman and Pyro in the X-Men film series. The one is An Ice Person and a Nice Guy; the other is an arrogant pyrokinetic who joins up with Magneto. Iceman wins when they finally go head-to-head. Literally.

Literature

A.L. Phillips's The Quest of the Unaligned has mages tied to a single element, using the classic four-element system. We see in the story that aeshes (fire) and shamais (water) don't get along, as the aesh heroine quarrels with a shamai the group meets and suspects him of stealing the Crown, and he in turn suspects her of being a traitor or just dangerously ambitious. Presumably, ruahks (wind) and aretzes (earth) also don't get along, but we only once see a ruahk and aretz in the same scene (though we do see that ruahks really don't like being underground.)

Also, ruahks apparently blame aretzes for hay fever.

In Ruth Frances Long's The Treachery of Beautiful Things, Jack is a creature of earth and air. To get back Jenny's soul from the nix, he must go into water. He is warned that this will mean he will not come back the same as he was. During the fight, the elemental oppositions are repeatedly mentioned.

Live-Action TV

The first Bad Future in Heroes has a fire-using Peter battling Sylar, who is using ice powers.

In Gotham's Season 2 finale, Firefly with her flamethrower and Fries with his liquid-helium gun go at it in a fire-vs-ice duel.

Music

In the Emerald Sword Saga, the bad guys are almost always associated with fire, while our heroes are marked with wind.

Mythology and Religion

Judeo-Christian religions generally associate God with Light and/or the Heavens, while the Satan character or God of Evil will be associated with Darkness and/or Earth. One the other hand God is the giver of restful shade while Satan appears as an angel of light and bright red dragon.

Leviathan (Sea) and Behemoth (Earth) from the same religious family. They are the most proud of beasts and will eventually kill each other.

In Egyptian Mythology we have Horus the Sun god opposed to Set, the god of storms and murderer of Horus' father Osiris.

In Eastern Mythology, there are the Demon-turned-gods Fujin (wind) and Raijin (thunder).

The four Elemental Lords of the Forgotten Realms setting each hate the one that embodies their opposite element. Akadi the Lady of Air battles Grumbar the Earthlord, and Kossuth the Firelord duels Istishia the Water Lord. This also holds true for their worshipers: according to the 3E supplement Faiths & Pantheons battles between Elemental Archons of opposing elements are the stuff of legends.

This can go even further because each of them can be (very roughly) thematically associated with one of the classic elements. Khorne is fire (rage and combat) and is opposed to Slannesh's water (sensuous, flowing). Tzeentch is air (which is often associated with intelligence, and is ever-changing) and stands opposed to Nurgle's earth (plodding, tough, enduring).

The former coaster at Universal's Islands of Adventure, Dueling Dragons, was about the conflict between two equally evil dragons, whom are literally named "Fire" and "Ice".

Toys

Most Toa teams in BIONICLE have a Toa of Fire as The Hero and a Toa of Ice as The Lancer, typically having a Red Oni, Blue Oni dynamic. Tahu and Kopaka from the Toa Mata and Toa Nuva arcs are the most prominent example, although they're both just as likely to start arguing with Gali, the Toa of Water, as they are with each other.

While not rivals in entirely, the Series 1 and Series 2 tribe of Mixels are based on contrasting elements and are often paired together, leading to interesting Mixes, and often embarrassing Murps. This includes the fire-based Infernites vs. the ice-based Frosticons, the electric-based Electroids vs. the rubber-based Flexers, and the earth-based Cragsters vs. the plant-based Fang Gang.

Video Games

In Pokémon your rival will always pick up the pokemon who has the elemental advantage over yours. Legendary duos tend to have opposing "elements":

Pokémon Diamond and Pearl also had Dialga who represents time, and Palkia who represents space. More subtly, they also represent Steel and Water, which are two of the best defensive types, and further Solid and Liquid (making a third Gas with Giratina).

One rivalry between two elements that's been prominent in Pokémon Black and White is the one between FireandElectricity. With Reshiram representing Fire and Yang, and Zekrom representing Electricity and Yin. This actually appeared earlier with trainers Flint and Volkner and perhaps even earlier still with Magmar and Electabuzz.

Samurai Warriors has Kenshin (ice) and Shingen (fire), Yoshihiro (wind) and Ginchiyo (thunder) and also Nobunaga (wind) and Mitsuhide (thunder). Note that in most cases it refers to the chosen element of their ultimate weapon.

Mitsunari (darkness) and Ieyasu (light) in Sengoku Basara. It also has the duo Kenshin (ice) and Shingen (fire) as tamer rivals.

Onimusha: Dawn of Dreams has the Genma God of Light (Fortinbras) opposed to the Oni God of Darkness (Soki).

Prince of Persia: The Forgotten Sands has the elemental opposition between Razia the Marid Queen (who has powers over water, which in the series symbolize life) and the Ifrit Lord Ratash, who has power over both fire and sand (which, in the series, is opposed to water as a symbol of death and entropy.)

Somewhat indirectly we have Raze and Ulrika, who got their powers from the Light Mana and Dark Mana respectively. Indirect because while Raze is a truly Light 'em Up guy, Ulrika is more like Soul Power. The real Casting a Shadow person is Ulrika's best friend, Chloe, who doesn't have much interaction with Raze otherwise.

There is also some sort of rivalry between Whim and Yun, being a Water Mana and Fire Mana respectively.

Tales of Xillia has Alvin's fire artes against Presa's ice and water artes, contrasting their personalities (though Alvin is rather laidback) and standing as part of the hero group and the Chimeriad. Also hints at a connection between the two; they used to date.

Kaguya Houraisan, representing the Moon, and Fujiwara no Mokou, representing Sun. While Kaguya's actual powers don't have much to do with the Moon, she is a Lunarian. Mokou meanwhile has a lot of The Phoenix theme going on.

According to the lore of The Elder Scrolls series, Flame Atronachs and Frost Atronachs are locked in a state of permanent warfare against each other all across the Oblivion planes. It takes the influence of a Daedric Prince like Mehrunes Dagon to make them stop fighting each other even temporarily.

In Ballad of Rapunzel, the seventh game in the Dark Parables series, it's explained that the nature goddess Flora appoints pairs of servants and grants them opposing elemental powers. The pairs are usually siblings; at least, all of the examples shown in the games thus far have been. The two most dramatic examples are the powers of life and death, which she gave to half-sisters Rapunzel and Belladonna, and the powers of ice and fire, which were given to Snow White and her twin brother Ross Red.

Western Animation

In Adventure Time Flame Princess (fire elemental) sees Finn crying over her. She thinks he's a water elemental because he "releases water," which is why they can't get along.

Avatar: The Last Airbender: One of the reasons it takes a while for Aang to get the hang of earthbending from Toph is because air's affinities (free-flowing, evasive) are diametrically opposed to earth's (direct and confrontational).

The Real Ghostbusters has an episode with two gods fighting over and causing havok motivated by an eternal elemental rivalry as one of the gods handle water and air and the other fire and earth.

Shadow Raiders is basically this with sci-fi versions of the elements. The different planets of the cluster; Water, Rock, Bone, Ice and Fire are enemies since the dawn of time.

The most obvious case in Winx Club is the fairy Bloom (Fire) who is the main heroine and the witch Icy (Ice) who is one of most recurring villain.

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